Efficient method for finding the number of combinations of values so that the sum is a certain number

Question

I know one can make a sample space for this problem, but are there any other ways of solving a problem like this:

Each letter has a number value. Find all possible combinations of the letters that sum of the number value is 15. Each letter can only be used once in a combination. Order doesn't matter, aik is the same as iak, kia, etc. a=5,b=2,c=3,d=3,e=3,f=1,g=2,h=2,i=9,j=1,k=1

An example of a correct combination: aik

An example of a incorrect combination: aacjk

Does anyone know a method that can be used to find the combinations for questions like this? I have a ti-89, so an easy method using it would be fine. Or an equation.

(sorry I don't know how to use the mathscript)

Answer 1

It seems that recursion is a natural and good algorithm for this problem. The input $(n; a_1,a_2,\dots a_k)$ of the algorithm consists of a number $n$ (15) and an ordered sequence of values $a_1\ge a_2\ge\dots\ge a_k$ (9,5,3,3,3,2,2,2,1,1,1). The output of the algorithm consists of the union of the outputs of the algorithm applied to inputs $(n-a_1;a_2,\dots a_k)$ and $(n;a_l,\dots a_k)$, where $l=\min\{m:a_m<a_1\}$.